9 MEASURABLE FUNCTIONS 147
12. (a) Let (//)o 5=^/1 be a sequence of real numbers, and let m be an integer fS/z. Let
Lm be the set of indices / with the following property : there exists an integer p such
that 0 fg;? <; w and tt + tl+1 -\ ----- h fi+P-i ^ 0. Show that Y tt *> 0. (Observe that
t€Lm
if i e Lm , and if /? is the smallest of the integers with the above property, then
i + 1, . . . , i -f p ~ 1 belong to Lm .)
(b) Let X be a locally compact space and p, a positive measure on X. Let u : X -> X
be a proper continuous mapping such that U(JJL) = p.. If /is any ^-integrable function
on X, put/o =/and/t =/o uk (k ]> 1). For each integer w, let Am be the measurable
set of points x e X such that one of the sums/0(jt) + /i(x) -f • • • + /P(x), where/? ^ w,
is ^> 0. Show that /(x) <tf/x(x) ^ 0. (For each integer n > 0 and each x e X, consider
jAm
the sequence (/((jc))o^^n+m , and apply (a) to this sequence, denoting the correspond-
ing set of indices by Lm(x). For each k <! n + m, let Bfc be the set of x e X such that
A: e Lm(x), and deduce from (a) that Y /A(x) du(x) ^> 0. Now use the fact that
* = 0JBk
Bft = u~k(Am) for 0 g A: g « to deduce that
(« + 1) f /W ^W + m f |/(*)| ^W ^ 0
JA»n J
for all n.) If A is the union of the Am , conclude that f(x) dp,(x) ^ 0 (" maximal
ergodic theorem").
(c) Let a be a real number and C an integrable set such that
1 "-1
a < lim sup - X /*(*)
n-*oo M fc = 0
for all x e C. Show that aju-(C) <; f |/(x)| dp(x). (Apply (b) to the function /- a<pc .)
(d) Let a, b be two real numbers such that a < b. If E is the set of points x e X
such that
I n-l ln~*
lim inf - ]£ /fc(x) < a < 6 < lim sup - V /*(*),
n-foo n * = 0 n-+oo W fc« 0
show that E is ^-negligible. (Deduce first from (c) that E is integrable, and then apply
(b) to each of the functions (/— b)<pE and (a -~~f)(pE, using the fact that w(E) <= E.)
Hence show that, for almost all XE X, the sequence
/I "-1 \
(-Z/(«*(*»)
\n*.o /
converges to a limit /*(x), that /* is integrable and that /*(w(x))=/*(x) almost
everywhere (G. D. Birkhoff's ergodic theorem). (Take a, b to be all pairs of rational
numbers such that a<b.)
(e) If X is compact, show that I /* d^ = fd^. (Reduce to the case /2> 0, and
consider first the case where/is bounded; then pass to the general case by observing to show that it implies that there exists a rational number r